By W. W. Rouse Ball

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Open Question 2 Let d ≥ 2. Assume given a RWRE for which (UE) and (IID) are fulfilled, and which is transient in direction l ∈ Sd−1 . Is the RWRE necessarily ballistic in direction l? As it is discussed above, Example 1 shows that if the hypothesis (UE) is replaced by (E) in the Open Question 2, then its answer is negative. The following proposition gives an indication of how much ellipticity should be required. d. environment. Assume that max E e∈U 1 = ∞. 1 − ω(0, e)ω(0, −e) (31) Then the walk is not ballistic in any direction.

D. environments, in dimensions d ≥ 2, directional transience implies ballisticity. The second chapter of these notes reviews this question as well as the progress and understanding which have been achieved toward its resolution. In particular, we introduce the fundamental concept of renewal times. We then proceed to the ballisticity conditions, under which it has been possible to obtain a better understanding of the so-called slowdown phenomena as well as of the ballistic and diffusive behavior in the setting of (uniformly) elliptic environments.

Rassoul-Agha in [32], obtains a version of Corollary 1 where transience is replaced by the so-called Kalikow’s condition [22], a stronger mixing assumption than ergodicity is required, but it is necessary only to assume the existence of an invariant probability measure which is absolutely continuous with respect to the initial law only on appropriate half-spaces. d. case and later extended by Alili [1] to the ergodic case. 40 A. Drewitz and A. F. Ramírez Theorem 4 Consider a RWRE in dimension d = 1 in an environment with law P fulfilling (E) and (ERG).

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A short account of the history of mathematics by W. W. Rouse Ball


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